INFINITE SEQUENCES AND SERIES 12 INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES Infinite sequences and series were introduced briefly in A Preview of Calculus in connection with Zeno’s paradoxes and the decimal representation of numbers.

INFINITE SEQUENCES AND SERIES Their importance in calculus stems from Newton’s idea of representing functions as sums of infinite series. For instance, in finding areas, he often integrated a function by first expressing it as a series and then integrating each term of the series.

INFINITE SEQUENCES AND SERIES We will pursue his idea in Section 12.10 in order to integrate such functions as e-x2. Recall that we have previously been unable to do this.

INFINITE SEQUENCES AND SERIES Many of the functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series. It is important to be familiar with the basic concepts of convergence of infinite sequences and series.

INFINITE SEQUENCES AND SERIES Physicists also use series in another way, as we will see in Section 12.11 In studying fields as diverse as optics, special relativity, and electromagnetism, they analyze phenomena by replacing a function with the first few terms in the series that represents it.

12.1 Sequences In this section, we will learn about: INFINITE SEQUENCES AND SERIES 12.1 Sequences In this section, we will learn about: Various concepts related to sequences.

SEQUENCE A sequence can be thought of as a list of numbers written in a definite order: a1, a2, a3, a4, …, an, … The number a1 is called the first term, a2 is the second term, and in general an is the nth term.

We will deal exclusively with infinite sequences. So, each term an will have a successor an+1.

So, a sequence can be defined as: SEQUENCES Notice that, for every positive integer n, there is a corresponding number an. So, a sequence can be defined as: A function whose domain is the set of positive integers

SEQUENCES However, we usually write an instead of the function notation f(n) for the value of the function at the number n.

The sequence { a1, a2, a3, . . .} is also denoted by: SEQUENCES Notation The sequence { a1, a2, a3, . . .} is also denoted by:

Some sequences can be defined by giving a formula for the nth term. Example 1 Some sequences can be defined by giving a formula for the nth term.

In the following examples, we give three descriptions of the sequence: SEQUENCES Example 1 In the following examples, we give three descriptions of the sequence: Using the preceding notation Using the defining formula Writing out the terms of the sequence

Preceding Notation Defining Formula Terms of Sequence SEQUENCES Example 1 a Preceding Notation Defining Formula Terms of Sequence In this and the subsequent examples, notice that n doesn’t have to start at 1.

Preceding Notation Defining Formula Terms of Sequence SEQUENCES Example 1 b Preceding Notation Defining Formula Terms of Sequence

Preceding Notation Defining Formula Terms of Sequence SEQUENCES Example 1 c Preceding Notation Defining Formula Terms of Sequence

Preceding Notation Defining Formula Terms of Sequence SEQUENCES Example 1 d Preceding Notation Defining Formula Terms of Sequence

Find a formula for the general term an of the sequence SEQUENCES Example 2 Find a formula for the general term an of the sequence assuming the pattern of the first few terms continues.

SEQUENCES Example 2 We are given that:

SEQUENCES Example 2 Notice that the numerators of these fractions start with 3 and increase by 1 whenever we go to the next term. The second term has numerator 4 and the third term has numerator 5. In general, the nth term will have numerator n+2.

The denominators are the powers of 5. SEQUENCES Example 2 The denominators are the powers of 5. Thus, an has denominator 5n.

The signs of the terms are alternately positive and negative. SEQUENCES Example 2 The signs of the terms are alternately positive and negative. Hence, we need to multiply by a power of –1.

Here, we want to start with a positive term. SEQUENCES Example 2 In Example 1 b, the factor (–1)n meant we started with a negative term. Here, we want to start with a positive term.

Thus, we use (–1)n–1 or (–1)n+1. SEQUENCES Example 2 Thus, we use (–1)n–1 or (–1)n+1. Therefore,

SEQUENCES Example 3 We now look at some sequences that don’t have a simple defining equation.

SEQUENCES Example 3 a The sequence {pn}, where pn is the population of the world as of January 1 in the year n

SEQUENCES Example 3 b If we let an be the digit in the nth decimal place of the number e, then {an} is a well-defined sequence whose first few terms are: {7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, …}

The Fibonacci sequence {fn} is defined recursively by the conditions Example 3 c The Fibonacci sequence {fn} is defined recursively by the conditions f1 = 1 f2 = 1 fn = fn–1 + fn–2 n ≥ 3 Each is the sum of the two preceding term terms. The first few terms are: {1, 1, 2, 3, 5, 8, 13, 21, …}

FIBONACCI SEQUENCE Example 3 This sequence arose when the 13th-century Italian mathematician Fibonacci solved a problem concerning the breeding of rabbits. See Exercise 71.

SEQUENCES A sequence such as that in Example 1 a [an = n/(n + 1)] can be pictured either by: Plotting its terms on a number line Plotting its graph Fig. 12.1.2, p. 712 Fig. 12.1.1, p. 712

SEQUENCES Note that, since a sequence is a function whose domain is the set of positive integers, its graph consists of isolated points with coordinates (1, a1) (2, a2) (3, a3) … (n, an) … Fig. 12.1.2, p. 712

SEQUENCES From either figure, it appears that the terms of the sequence an = n/(n + 1) are approaching 1 as n becomes large. Fig. 12.1.2, p. 712 Fig. 12.1.1, p. 712

SEQUENCES In fact, the difference can be made as small as we like by taking n sufficiently large. We indicate this by writing

SEQUENCES In general, the notation means that the terms of the sequence {an} approach L as n becomes large.

SEQUENCES Notice that the following definition of the limit of a sequence is very similar to the definition of a limit of a function at infinity given in Section 2.6

LIMIT OF A SEQUENCE Definition 1 A sequence {an} has the limit L, and we write if we can make the terms an as close to L as we like, by taking n sufficiently large. If exists, the sequence converges (or is convergent). Otherwise, it diverges (or is divergent).

LIMIT OF A SEQUENCE Here, Definition 1 is illustrated by showing the graphs of two sequences that have the limit L. Fig. 12.1.3, p. 713

A more precise version of Definition 1 is as follows. LIMIT OF A SEQUENCE A more precise version of Definition 1 is as follows.

A sequence {an} has the limit L, and we write LIMIT OF A SEQUENCE Definition 2 A sequence {an} has the limit L, and we write if for every ε > 0 there is a corresponding integer N such that: if n > N then |an – L| < ε

LIMIT OF A SEQUENCE Definition 2 is illustrated by the figure, in which the terms a1, a2, a3, . . . are plotted on a number line. Fig. 12.1.4, p. 713

LIMIT OF A SEQUENCE No matter how small an interval (L – ε, L + ε) is chosen, there exists an N such that all terms of the sequence from aN+1 onward must lie in that interval. Fig. 12.1.4, p. 713

Another illustration of Definition 2 is given here. LIMIT OF A SEQUENCE Another illustration of Definition 2 is given here. The points on the graph of {an} must lie between the horizontal lines y = L + ε and y = L – ε if n > N. Fig. 12.1.5, p. 713

This picture must be valid no matter how small ε is chosen. LIMIT OF A SEQUENCE This picture must be valid no matter how small ε is chosen. Usually, however, a smaller ε requires a larger N. Fig. 12.1.5, p. 713

LIMITS OF SEQUENCES If you compare Definition 2 with Definition 7 in Section 2.6, you will see that the only difference between and is that n is required to be an integer. Thus, we have the following theorem.

If and f(n) = an when n is an integer, then LIMITS OF SEQUENCES Theorem 3 If and f(n) = an when n is an integer, then

Theorem 3 is illustrated here. LIMITS OF SEQUENCES Theorem 3 is illustrated here. Fig. 12.1.6, p. 714

LIMITS OF SEQUENCES Equation 4 In particular, since we know that when r > 0 (Theorem 5 in Section 2.6), we have:

If an becomes large as n becomes large, we use the notation LIMITS OF SEQUENCES If an becomes large as n becomes large, we use the notation The following precise definition is similar to Definition 9 in Section 2.6

LIMIT OF A SEQUENCE Definition 5 means that, for every positive number M, there is an integer N such that if n > N then an > M

If , then the sequence {an} is divergent, but in a special way. LIMITS OF SEQUENCES If , then the sequence {an} is divergent, but in a special way. We say that {an} diverges to ∞.

LIMITS OF SEQUENCES The Limit Laws given in Section 2.3 also hold for the limits of sequences and their proofs are similar.

LIMIT LAWS FOR SEQUENCES Suppose {an} and {bn} are convergent sequences and c is a constant.

LIMIT LAWS FOR SEQUENCES Then,

LIMIT LAWS FOR SEQUENCES Also,

The Squeeze Theorem can also be adapted for sequences, as follows. LIMITS OF SEQUENCES The Squeeze Theorem can also be adapted for sequences, as follows.

SQUEEZE THEOREM FOR SEQUENCES If an ≤ bn ≤ cn for n ≥ n0 and , then Fig. 12.1.7, p. 715

LIMITS OF SEQUENCES Another useful fact about limits of sequences is given by the following theorem. The proof is left as Exercise 75.

LIMITS OF SEQUENCES Theorem 6 If then

Find The method is similar to the one we used in Section 2.6 LIMITS OF SEQUENCES Example 4 Find The method is similar to the one we used in Section 2.6 We divide the numerator and denominator by the highest power of n and then use the Limit Laws.

Thus, Here, we used Equation 4 with r = 1. LIMITS OF SEQUENCES Example 4 Thus, Here, we used Equation 4 with r = 1.

LIMITS OF SEQUENCES Example 5 Calculate Notice that both the numerator and denominator approach infinity as n → ∞.

Here, we can’t apply l’Hospital’s Rule directly. LIMITS OF SEQUENCES Example 5 Here, we can’t apply l’Hospital’s Rule directly. It applies not to sequences but to functions of a real variable.

LIMITS OF SEQUENCES Example 5 However, we can apply l’Hospital’s Rule to the related function f(x) = (ln x)/x and obtain:

Therefore, by Theorem 3, we have: LIMITS OF SEQUENCES Example 5 Therefore, by Theorem 3, we have:

Determine whether the sequence an = (–1)n is convergent or divergent. LIMITS OF SEQUENCES Example 6 Determine whether the sequence an = (–1)n is convergent or divergent.

LIMITS OF SEQUENCES Example 6 If we write out the terms of the sequence, we obtain: {–1, 1, –1, 1, –1, 1, –1, …}

The graph of the sequence is shown. LIMITS OF SEQUENCES Example 6 The graph of the sequence is shown. The terms oscillate between 1 and –1 infinitely often. Thus, an does not approach any number. Fig. 12.1.8, p. 715

That is, the sequence {(–1)n} is divergent. LIMITS OF SEQUENCES Example 6 Thus, does not exist. That is, the sequence {(–1)n} is divergent.

Evaluate if it exists. Thus, by Theorem 6, LIMITS OF SEQUENCES Example 7 Evaluate if it exists. Thus, by Theorem 6, Fig. 12.1.9, p. 716

LIMITS OF SEQUENCES The following theorem says that, if we apply a continuous function to the terms of a convergent sequence, the result is also convergent.

If and the function f is continuous at L, then LIMITS OF SEQUENCES Theorem 7 If and the function f is continuous at L, then The proof is left as Exercise 76.

Find The sine function is continuous at 0. LIMITS OF SEQUENCES Example 8 Find The sine function is continuous at 0. Thus, Theorem 7 enables us to write:

LIMITS OF SEQUENCES Example 9 Discuss the convergence of the sequence an = n!/nn, where n! = 1 . 2 . 3 . … . n

Both the numerator and denominator approach infinity as n → ∞. LIMITS OF SEQUENCES Example 9 Both the numerator and denominator approach infinity as n → ∞. However, here, we have no corresponding function for use with l’Hospital’s Rule. x! is not defined when x is not an integer.

LIMITS OF SEQUENCES Example 9 Let’s write out a few terms to get a feeling for what happens to an as n gets large:

LIMITS OF SEQUENCES E. g. 9—Equation 8 Therefore,

LIMITS OF SEQUENCES Example 9 From these expressions and the graph here, it appears that the terms are decreasing and perhaps approach 0. Fig. 12.1.10, p. 716

To confirm this, observe from Equation 8 that LIMITS OF SEQUENCES Example 9 To confirm this, observe from Equation 8 that Notice that the expression in parentheses is at most 1 because the numerator is less than (or equal to) the denominator.

Thus, 0 < an ≤ We know that 1/n → 0 as n → ∞. LIMITS OF SEQUENCES Example 9 Thus, 0 < an ≤ We know that 1/n → 0 as n → ∞. Therefore an → 0 as n → ∞ by the Squeeze Theorem.

For what values of r is the sequence {r n} convergent? LIMITS OF SEQUENCES Example 10 For what values of r is the sequence {r n} convergent? From Section 2.6 and the graphs of the exponential functions in Section 1.5, we know that for a > 1 and for 0 < a < 1.

Thus, putting a = r and using Theorem 3, we have: LIMITS OF SEQUENCES Example 10 Thus, putting a = r and using Theorem 3, we have: It is obvious that

If –1 < r < 0, then 0 < |r | < 1. LIMITS OF SEQUENCES Example 10 If –1 < r < 0, then 0 < |r | < 1. Thus, Therefore, by Theorem 6,

If r ≤ –1, then {r n} diverges as in Example 6. LIMITS OF SEQUENCES Example 10 If r ≤ –1, then {r n} diverges as in Example 6. Fig. 12.1.11, p. 717

The figure shows the graphs for various values of r. LIMITS OF SEQUENCES Example 10 The figure shows the graphs for various values of r. Fig. 12.1.11, p. 717

The case r = –1 was shown earlier. LIMITS OF SEQUENCES Example 10 The case r = –1 was shown earlier. Fig. 12.1.8, p. 715

The results of Example 10 are summarized for future use, as follows. LIMITS OF SEQUENCES The results of Example 10 are summarized for future use, as follows.

LIMITS OF SEQUENCES Equation 9 The sequence {r n} is convergent if –1 < r ≤ 1 and divergent for all other values of r.

A sequence {an} is called: LIMITS OF SEQUENCES Definition 10 A sequence {an} is called: Increasing, if an < an+1 for all n ≥ 1, that is, a1 < a2 < a3 < · · · Decreasing, if an > an+1 for all n ≥ 1 Monotonic, if it is either increasing or decreasing

The sequence is decreasing because DECREASING SEQUENCES Example 11 The sequence is decreasing because and so an > an+1 for all n ≥ 1.

Show that the sequence is decreasing. DECREASING SEQUENCES Example 12 Show that the sequence is decreasing.

We must show that an+1 < an, that is, DECREASING SEQUENCES E. g. 12—Solution 1 We must show that an+1 < an, that is,

DECREASING SEQUENCES E. g. 12—Solution 1 This inequality is equivalent to the one we get by cross-multiplication:

Since n ≥ 1, we know that the inequality n2 + n > 1 is true. DECREASING SEQUENCES E. g. 12—Solution 1 Since n ≥ 1, we know that the inequality n2 + n > 1 is true. Therefore, an+1 < an. Hence, {an} is decreasing.

DECREASING SEQUENCES E. g. 12—Solution 2 Consider the function

Thus, f is decreasing on (1, ∞). DECREASING SEQUENCES E. g. 12—Solution 2 Thus, f is decreasing on (1, ∞). Hence, f(n) > f(n + 1). Therefore, {an} is decreasing.

A sequence {an} is bounded: BOUNDED SEQUENCES Definition 11 A sequence {an} is bounded: Above, if there is a number M such that an ≤ M for all n ≥ 1 Below, if there is a number m such that m ≤ an for all n ≥ 1 If it is bounded above and below

BOUNDED SEQUENCES For instance, The sequence an = n is bounded below (an > 0) but not above. The sequence an = n/(n+1) is bounded because 0 < an < 1 for all n.

We know that not every bounded sequence is convergent. BOUNDED SEQUENCES We know that not every bounded sequence is convergent. For instance, the sequence an = (–1)n satisfies –1 ≤ an ≤ 1 but is divergent from Example 6.

Similarly, not every monotonic sequence is convergent (an = n → ∞). BOUNDED SEQUENCES Similarly, not every monotonic sequence is convergent (an = n → ∞).

BOUNDED SEQUENCES However, if a sequence is both bounded and monotonic, then it must be convergent. This fact is proved as Theorem 12. However, intuitively, you can understand why it is true by looking at the following figure.

BOUNDED SEQUENCES If {an} is increasing and an ≤ M for all n, then the terms are forced to crowd together and approach some number L. Fig. 12.1.12, p. 718

BOUNDED SEQUENCES The proof of Theorem 12 is based on the Completeness Axiom for the set of real numbers. This states that, if S is a nonempty set of real numbers that has an upper bound M (x ≤ M for all x in S), then S has a least upper bound b.

This means: b is an upper bound for S. BOUNDED SEQUENCES This means: b is an upper bound for S. However, if M is any other upper bound, then b ≤ M.

BOUNDED SEQUENCES The Completeness Axiom is an expression of the fact that there is no gap or hole in the real number line.

Every bounded, monotonic sequence is convergent. MONOTONIC SEQ. THEOREM Theorem 12 Every bounded, monotonic sequence is convergent.

Suppose {an} is an increasing sequence. MONOTONIC SEQ. THEOREM Theorem 12—Proof Suppose {an} is an increasing sequence. Since {an} is bounded, the set S = {an|n ≥ 1} has an upper bound. By the Completeness Axiom, it has a least upper bound L.

MONOTONIC SEQ. THEOREM Theorem 12—Proof Given ε > 0, L – ε is not an upper bound for S (since L is the least upper bound). Therefore, aN > L – ε for some integer N

However, the sequence is increasing. So, an ≥ aN for every n > N. MONOTONIC SEQ. THEOREM Theorem 12—Proof However, the sequence is increasing. So, an ≥ aN for every n > N. Thus, if n > N, we have an > L – ε Since an ≤ L, thus 0 ≤ L – an < ε

Thus, |L – an| < ε whenever n > N Therefore, MONOTONIC SEQ. THEOREM Theorem 12—Proof Thus, |L – an| < ε whenever n > N Therefore,

MONOTONIC SEQ. THEOREM Theorem 12—Proof A similar proof (using the greatest lower bound) works if {an} is decreasing.

MONOTONIC SEQ. THEOREM The proof of Theorem 12 shows that a sequence that is increasing and bounded above is convergent. Likewise, a decreasing sequence that is bounded below is convergent.

This fact is used many times in dealing with infinite series. MONOTONIC SEQ. THEOREM This fact is used many times in dealing with infinite series.

Investigate the sequence {an} defined by the recurrence relation MONOTONIC SEQ. THEOREM Example 13 Investigate the sequence {an} defined by the recurrence relation

We begin by computing the first several terms: MONOTONIC SEQ. THEOREM Example 13 We begin by computing the first several terms: These initial terms suggest the sequence is increasing and the terms are approaching 6.

MONOTONIC SEQ. THEOREM Example 13 To confirm that the sequence is increasing, we use mathematical induction to show that an+1 > an for all n ≥ 1. Mathematical induction is often used in dealing with recursive sequences.

That is true for n = 1 because a2 = 4 > a1. MONOTONIC SEQ. THEOREM Example 13 That is true for n = 1 because a2 = 4 > a1.

If we assume that it is true for n = k, we have: MONOTONIC SEQ. THEOREM Example 13 If we assume that it is true for n = k, we have: Hence, and Thus,

We have deduced that an+1 > an is true for n = k + 1. MONOTONIC SEQ. THEOREM Example 13 We have deduced that an+1 > an is true for n = k + 1. Therefore, the inequality is true for all n by induction.

MONOTONIC SEQ. THEOREM Example 13 Next, we verify that {an} is bounded by showing that an < 6 for all n. Since the sequence is increasing, we already know that it has a lower bound: an ≥ a1 = 2 for all n

So, the assertion is true for n = 1. MONOTONIC SEQ. THEOREM Example 13 We know that a1 < 6. So, the assertion is true for n = 1.

Suppose it is true for n = k. MONOTONIC SEQ. THEOREM Example 13 Suppose it is true for n = k. Then, Thus, and Hence,

This shows, by mathematical induction, that an < 6 for all n. MONOTONIC SEQ. THEOREM Example 13 This shows, by mathematical induction, that an < 6 for all n.

MONOTONIC SEQ. THEOREM Example 13 Since the sequence {an} is increasing and bounded, Theorem 12 guarantees that it has a limit. However, the theorem doesn’t tell us what the value of the limit is.

MONOTONIC SEQ. THEOREM Example 13 Nevertheless, now that we know exists, we can use the recurrence relation to write:

Since an → L, it follows that an+1 → L, too (as n → ∞, n + 1 → ∞ too). MONOTONIC SEQ. THEOREM Example 13 Since an → L, it follows that an+1 → L, too (as n → ∞, n + 1 → ∞ too). Thus, we have: Solving this equation for L, we get L = 6, as predicted.